Isye 2027

# 5 p x k p x k 05 gaussian approximation with

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: entered at zero), shrinking it by a factor σ horizontally and stretching it by a factor σ vertically. Let’s check that the N (µ, σ 2 ) density indeed integrates to one, has mean µ, and variance σ 2 . To show that the normal density integrates to one, it suﬃces to check that the standard normal density integrates to one, because the density for general µ and σ 2 is obtained from the standard normal 2 ∞ pdf by the scaling rule, which preserves the total integral of the density. Let I = −∞ e−u /2 du. Then, switching to polar coordinates, we have: ∞ 2 /2 e−u I2 = −∞ ∞ ∞ −∞ 2π ∞ du −∞ ∞ 2 +v 2 )/2 e−(u e−r = 0 ∞ = 2π 2 /2 dv −∞ = 0 e −v e −r 0 = −2πe−r 2 /2 2 /2 2 /2 dudv rdrdθ rdr ∞ = 2π. 0 1 http://www.stat.tamu.edu/∼west/applets/normaldemo.html for example. 3.6. LINEAR SCALING OF PDFS AND THE GAUSSIAN DISTRIBUTION 91 √ Therefore, I = 2π, which means the standard normal density integrates to one, as claimed. The fact that µ is the mean of the N (µ, σ 2 ) density follows from the fact that...
View Full Document

Ask a homework question - tutors are online